Finite Group Databasebeta

← #22598438

Subgroups — #22598438

Order 9088.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986554C211yesmaximal, characteristic, complemented
2C4×D4×D71C211yesmaximal, characteristic
3#17986333C212yesmaximal, complemented
4#17986280C211yesmaximal, characteristic, complemented
5#17986279C211yesmaximal, characteristic, complemented
6#17986201C211yesmaximal, characteristic, complemented
7#17986148C212yesmaximal
8#17986102C212yesmaximal, complemented
9#17986100C212yesmaximal, complemented
10#17986055C211yesmaximal, characteristic, complemented
11#461×C71C211yesmaximal, characteristic, complemented
12C2×D4×D71C2211yescharacteristic
13C22×C4×D71—21no—
14C2×#11765852C2212yes—
15#14556128C2211yescharacteristic, complemented
16#14556119—21no—
17#14556108—21no—
18#14556104C2212yes—
19Q8×Dic71C2212yescomplemented
20#14556093—21no—
21C4×D284C2211yescharacteristic
22D4×Dic71C2211yescharacteristic
23#14556090C2211yescharacteristic
24C4×C22⋊D71—21no—
25(C22⋊D71)⋊C4—21no—
26#14556086—21no—
27C4⋊C4×D71C2211yescharacteristic
28C22⋊C4×D71—21no—
29C42×D71C2211yescharacteristic
30#14556062C2212yescomplemented
31#14556061C2212yescomplemented
32C4⋊(C71⋊C8)C2211yescharacteristic, complemented
33#14556052C2212yescomplemented
34#14556050C2212yes—
35#14556049C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556038C2212yescomplemented
38C2×SD16×C71C2212yes—
39C4⋊Q8×C71C2211yescharacteristic, complemented
40C4×D4×C71C2211yescharacteristic
41C4⋊C8×C71C2211yescharacteristic, complemented
42#103×C71C2212yescomplemented
43C23×D71—21no—
44C2×Dic142—21no—
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—21no—
47D4×D71—22no—
48D4×D71—22no—
49C22×Dic71—21no—
50C2×C4×D71—21no—
51C2×C4×D71—21no—
52C2×C4×D71—21no—
53C2×C4×D71—44no—
54C2×C4×D71D411yescharacteristic
55C2×C4×D71D411yescharacteristic
56C2×C4×D71C2311yescharacteristic
57#11765852—24no—
58C2×C71⋊C8C2312yes—
59C4⋊Dic71—21no—
60C4⋊Dic71—22no—
61C4⋊Dic71C2312yes—
62C4⋊Dic71C2311yescharacteristic
63Dic71⋊C4—21no—
64Dic71⋊C4—22no—
65C22⋊Dic71—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—21no—
68(C2×D71)⋊C4—21no—
69C4×Dic71—22no—
70C4×Dic71D411yescharacteristic
71C4×Dic71D411yescharacteristic
72C4×Dic71C2311yescharacteristic
73C2×Q8×C71C2312yes—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76SD16×C71—24no—
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71—21no—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C42×C71C2311yescharacteristic
83C23×C71—21no—
84C22×D71—24no—
85C22×D71—21no—
86C22×D71—42no—
87C22×D71—42no—
88C22×D71C2×D411yescharacteristic
89Q8×C71—22no—
90Q8×C71—22no—
91D4×C71—21no—
92D4×C71#4112yes—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—42no—
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—24no—
100C2×Dic71—22no—
101C2×Dic71—21no—
102C2×Dic71—42no—
103C2×Dic71C2×D411yescharacteristic
104C2×Dic71C2×D411yescharacteristic
105C2×Dic71C2×D411yescharacteristic
106D284—21no—
107D284#4112yes—
108C4×D71—22no—
109C4×D71—22no—
110C4×D71—21no—
111C4×D71—21no—
112C4×D71—42no—
113C4×D71—42no—
114C4×D71—42no—
115Dic142—42no—
116C8×C71—22no—
117C71⋊C8—22no—
118C22×C71—22no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71C22×D411yescharacteristic
122C2×D71—21no—
123C2×D71—21no—
124C2×D71—22no—
125C2×D71—48no—
126C2×D71—41no—
127C2×D71—41no—
128C2×D71#13712yes—
129C4×C71—21no—
130C4×C71—42no—
131C4×C71—41no—
132C4×C71AGL(1,ℤ/8)12yes—
133C4×C71C2×#4111yescharacteristic
134C4×C71C2×#4111yescharacteristic
135Dic71—21no—
136Dic71—21no—
137Dic71—21no—
138Dic71—42no—
139Dic71—41no—
140C2×C71—42no—
141C2×C71#52511yescharacteristic
142C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C71C2×#13711yescharacteristic
144D71—22no—
145D71—42no—
146#3110—711nomaximal, Sylow
147C71#311011yescharacteristic, complemented, Sylow
148#533—711no—
149C2×C4×D4—711no—
150#484—712no—
151#474—711no—
152#474—711no—
153#461—711no—
154#461D7111yescharacteristic, complemented
155#450—712no—
156#440—712no—
157#440—712no—
158#432—711no—
159C22×D4—711no—
160C23×C4—1421no—
161C2×SD16—712no—
162C2×SD16C2×D7112yes—
163C4⋊Q8—711no—
164C4⋊Q8C2×D7111yescharacteristic, complemented
165#125—1421no—
166C22⋊Q8—712no—
167C22⋊Q8—1421no—
168C4×Q8—712no—
169C4×Q8—1421no—
170C4×D4—711no—
171C4×D4—711no—
172C4×D4—711no—
173C4×D4—1421no—
174C4×D4—1421no—
175C4×D4C2×D7111yescharacteristic
176#117—1421no—
177C2×C4⋊C4—711no—
178C2×C22⋊C4—1421no—
179C2×C42—711no—
180#107—712no—
181#107—712no—
182C4⋊C8—711no—
183C4⋊C8C2×D7111yescharacteristic, complemented
184#103—712no—
185#103C2×D7112yescomplemented
186#102—712no—
187#102—712no—
188C22⋊C8—712no—
189#97—712no—
190C24—1421no—
191C2×Q8—1421no—
192C2×Q8C22×D7112yes—
193C2×D4—711no—
194C2×D4—1421no—
195C2×D4—1422no—
196C2×D4—1422no—
197C2×D4C22×D7111yescharacteristic
198C22×C4—21no—
199C22×C4—2844no—
200C22×C4—711no—
201C22×C4—711no—
202C22×C4—711no—
203C22×C4—1421no—
204C22×C4—1421no—
205C22×C4—1421no—
206C22×C4—1421no—
207SD16—24no—
208SD16—1424no—
209C2×C8—712no—
210C2×C8C22×D7112yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—712no—
214C4⋊C4—711no—
215C4⋊C4—1421no—
216C4⋊C4—1422no—
217C4⋊C4—1422no—
218C4⋊C4—1421no—
219C4⋊C4C22×D7111yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—1421no—
222C22⋊C4—1422no—
223C22⋊C4—1421no—
224C22⋊C4—1421no—
225C42—711no—
226C42—711no—
227C42—711no—
228C42—1422no—
229C42C22×D7111yescharacteristic
230C23—21no—
231C23—2842no—
232C23—2842no—
233C23—711no—
234C23—1424no—
235C23—1421no—
236Q8—22no—
237Q8—22no—
238Q8—2842no—
239D4—21no—
240D4—712no—
241D4—1424no—
242D4—1421no—
243D4#1176586412yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—2842no—
248C2×C4—2842no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4—711no—
252C2×C4—711no—
253C2×C4—711no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4—1421no—
257C2×C4—1421no—
258C2×C4—1421no—
259C2×C4—1422no—
260C2×C4D4×D7111yescharacteristic
261C2×C4D4×D7111yescharacteristic
262C2×C4C23×D7111yescharacteristic
263C8—22no—
264C8—1422no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—2848no—
269C22—2841no—
270C22—2841no—
271C22—712no—
272C22—1422no—
273C22—1421no—
274C22—1421no—
275C22C2×D4×D7111yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—2841no—
280C4—2842no—
281C4—1421no—
282C4—1421no—
283C4—1421no—
284C4#1455615512yes—
285C4#41×D7111yescharacteristic
286C4C2×#1176586411yescharacteristic
287C2—42no—
288C2—2842no—
289C2—1422no—
290C2#1798650911yescharacteristic
291C2C2×#1455615511yescharacteristic
292C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#461×C71C2
socleC22×C71C22×D4
hypercenter#461D71
hypocenterC71#3110
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986333C2semidirect
4. #17986280C2semidirect
5. #17986279C2semidirect
6. #17986201C2semidirect
8. #17986102C2semidirect
9. #17986100C2semidirect
10. #17986055C2semidirect
11. #461×C71C2semidirect
15. #14556128C22semidirect
19. Q8×Dic71C22semidirect
30. #14556062C22semidirect
31. #14556061C22semidirect
32. C4⋊(C71⋊C8)C22semidirect
33. #14556052C22semidirect
37. #14556038C22semidirect
39. C4⋊Q8×C71C22semidirect
41. C4⋊C8×C71C22semidirect
42. #103×C71C22semidirect
147. C71#3110semidirect
154. #461D71semidirect
164. C4⋊Q8C2×D71semidirect
183. C4⋊C8C2×D71semidirect
185. #103C2×D71semidirect